`|(a xx b).c | = |a||b||c|` , if

Options

A. a.b = b.c = 0
B. b.c = c.a = 0
C. c.a = a.b = 0
D. a.b = b.c = c.a = 0

Answer

To solve the problem \( |(a \times b) \cdot c| = |a||b||c| \), we will analyze the conditions under which this equation holds true.

### Step-by-Step Solution:

1. **Understanding the Left Side**:
The left side of the equation is \( |(a \times b) \cdot c| \). The expression \( a \times b \) gives a vector that is perpendicular to both \( a \) and \( b \). The dot product \( (a \times b) \cdot c \) measures how much of vector \( c \) lies in the direction of the vector \( a \times b \).

2. **Magnitude of Cross Product**:
The magnitude of the cross product \( |a \times b| \) is given by:
\[ |a \times b| = |a||b| \sin \theta \]
where \( \theta \) is the angle between vectors \( a \) and \( b \).

3. **Dot Product with Vector \( c \)**:
Now, we can express the left side as:
\[ |(a \times b) \cdot c| = |a \times b| |c| \cos \alpha \]
where \( \alpha \) is the angle between the vector \( c \) and the vector \( a \times b \).

4. **Substituting the Magnitude of the Cross Product**:
Substituting the expression for \( |a \times b| \) into the equation gives:
\[ |(a \times b) \cdot c| = |a||b| \sin \theta |c| \cos \alpha \]

5. **Setting Up the Equation**:
We want this to equal \( |a||b||c| \):
\[ |a||b| \sin \theta |c| \cos \alpha = |a||b||c| \]

6. **Dividing Both Sides**:
Assuming \( |a|, |b|, |c| \neq 0 \), we can divide both sides by \( |a||b||c| \):
\[ \sin \theta \cos \alpha = 1 \]

7. **Analyzing the Equation**:
The equation \( \sin \theta \cos \alpha = 1 \) implies that both \( \sin \theta \) and \( \cos \alpha \) must be at their maximum values. This occurs when:
- \( \sin \theta = 1 \) (which means \( \theta = \frac{\pi}{2} \), or \( a \) and \( b \) are perpendicular)
- \( \cos \alpha = 1 \) (which means \( \alpha = 0 \), or \( c \) is in the same direction as \( a \times b \))

8. **Conclusion**:
From the above conditions, we conclude:
- If \( a \) and \( b \) are perpendicular, then \( a \cdot b = 0 \).
- If \( c \) is in the same direction as \( a \times b \), then \( c \) is perpendicular to both \( a \) and \( b \), which implies \( c \cdot a = 0 \) and \( c \cdot b = 0 \).

Thus, the conditions under which \( |(a \times b) \cdot c| = |a||b||c| \) hold true are:
\[ a \cdot b = 0, \quad c \cdot a = 0, \quad c \cdot b = 0 \]